1. Locate the absolute extrema of the function f(x)=cos(pi*x) on the closed interval [0,1/2].
2. Determine whether Rolle's Theorem applied to the function f(x)=x^2+6x+8 on the closed interval[-4,-2]. If Rolle's Theorem can be applied, find all values of c in the open interval (-4,-2) such that f'(c)=0.
3. Determine whether the open intervals on which the graph of f(x)=-7x+7cosx is concave upward or downward.
4. Find the points of inflection and discuss the concavity of the function f(x)=sinx-cosx on the interval (0,2pi)
5.Find the points of inflection and discuss the concavity of the function f(x)=-x^3+x^2-6x-5
1.
f' = -pi sin(pi*x) extrema where f' = 0, or x an integer
2.
since f(x) = (x+4)(x+2) f(-4)=f(-2)=0, so we're good to do. vertex is at x = -3.
3.
f is concave up if f'' > 0
f'' = -7cosx, so where is that >0? <0?
4.
concavity as above, inflection where f'' = 0
f'' = -sinx + cosx = √2 sin(x + π/4)
5.
same methods as in #3,4/
f'' = -6x
1. To locate the absolute extrema of the function f(x) = cos(pi*x) on the closed interval [0, 1/2], we can follow these steps:
- First, find the critical points of the function within the given interval by taking the derivative of f(x) and setting it equal to zero:
f'(x) = -pi*sin(pi*x) = 0
Setting this equation equal to zero and solving for x will give us the critical point(s).
- Next, evaluate the function at the critical points obtained to find the corresponding y-values.
- Finally, compare the y-values obtained from the critical points with the y-values at the endpoints of the interval (0 and 1/2) to determine which ones are the absolute extrema.
2. To determine whether Rolle's Theorem applied to the function f(x) = x^2 + 6x + 8 on the closed interval [-4, -2], we need to check if the conditions of Rolle's Theorem are met. The conditions are as follows:
- The function must be continuous on the closed interval [-4, -2].
- The function must be differentiable on the open interval (-4, -2).
- The function's values at the endpoints (-4 and -2) must be equal.
If all these conditions are met, then we can apply Rolle's Theorem. To find the values of c in the open interval (-4, -2) where f'(c) = 0, we need to take the derivative of f(x), set it equal to zero, and solve for x.
3. To determine whether the graph of f(x) = -7x + 7cos(x) is concave upward or downward on its open intervals, we need to examine the second derivative of the function. The concavity of a function is determined by the sign of its second derivative:
- If f''(x) > 0, then the graph is concave upward on the given interval.
- If f''(x) < 0, then the graph is concave downward on the given interval.
- If f''(x) = 0, then the graph may have points of inflection, and further analysis is needed.
To find the second derivative f''(x), first, take the derivative of f(x) with respect to x to get f'(x), and then take the derivative of f'(x) to get f''(x). Analyze the sign of f''(x) on the given interval(s) to determine the concavity.
4. To find the points of inflection and discuss the concavity of the function f(x) = sin(x) - cos(x) on the interval (0, 2pi), follow these steps:
- Take the second derivative of the function f(x) to get f''(x).
- Set f''(x) equal to zero and solve for x to find the potential points of inflection.
- Test the concavity of the function on each interval that is divided by the potential points of inflection using arbitrary test points.
- Discuss the concavity in each interval (concave upward or downward) and identify the exact points of inflection based on the sign changes in the concavity.
5. To find the points of inflection and discuss the concavity of the function f(x) = -x^3 + x^2 - 6x - 5, we can follow these steps:
- Take the second derivative of the function f(x) to get f''(x).
- Set f''(x) equal to zero and solve for x to find the potential points of inflection.
- Test the concavity of the function on each interval that is divided by the potential points of inflection using arbitrary test points.
- Discuss the concavity in each interval (concave upward or downward) and identify the exact points of inflection based on the sign changes in the concavity.